Finite Group Databasebeta

← #27320602

Subgroups — #27320602

Order 13568.

1694 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810526C211yesmaximal, characteristic
2C2×#16810517C212yesmaximal, complemented
3C2×#16810468C211yesmaximal, characteristic, complemented
4#20272979C211yesmaximal, characteristic, complemented
5#20272972C211yesmaximal, characteristic, complemented
6#20272964C212yesmaximal, complemented
7#20272958C212yesmaximal, complemented
8#20272956C211yesmaximal, characteristic
9#20272952C212yesmaximal, complemented
10#20272939C211yesmaximal, characteristic, complemented
11C53×#1438C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21#16810526—24no—
22#16810517—28no—
23#16810468—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C23×D4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2212yes—
35C22×C22⋊C4×C53C2212yes—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—42no—
46C2×#11719919—24no—
47C2×#11719919—42no—
48C2×#11719919—42no—
49C2×C22⋊Dic53—24no—
50C2×C22⋊Dic53—24no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—24no—
57C2×C22⋊Dic53—24no—
58C2×C22⋊Dic53—42no—
59C2×C22⋊Dic53—42no—
60C2×C22⋊Dic53—42no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68C2×C4×Dic53D414yes—
69C2×C4×Dic53D414yescomplemented
70#13133124—22no—
71#13133124—22no—
72#13133124—24no—
73#13133124—24no—
74#13133124—22no—
75#13133124—22no—
76#13133124—24no—
77#13133124—42no—
78#13133124—42no—
79C25×C53C2312yes—
80C22×D4×C53—24no—
81C22×D4×C53—22no—
82C22×D4×C53—22no—
83C22×D4×C53—22no—
84C22×D4×C53—42no—
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—24no—
91C2×C22⋊C4×C53—24no—
92C2×C22⋊C4×C53—42no—
93C2×C22⋊C4×C53—42no—
94C2×C22⋊C4×C53—42no—
95#95×C53—22no—
96#95×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53#4114yes—
112C24×C53C2411yescharacteristic
113C2×D4×C53—28no—
114C2×D4×C53—48no—
115C2×D4×C53—48no—
116C2×D4×C53—44no—
117C2×D4×C53—44no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D412yes—
171C22×Dic53C2×D412yes—
172C22×Dic53C2×D412yes—
173C22×Dic53#4114yes—
174C22×Dic53#4112yes—
175C22×Dic53#4112yes—
176C22⋊C4×C53—48no—
177C22⋊C4×C53—48no—
178C22⋊Dic53—216no—
179C22⋊Dic53—48no—
180C22⋊Dic53—48no—
181C22⋊Dic53—48no—
182C4⋊Dic53—44no—
183#11719919—48no—
184C4×Dic53—28no—
185C4×Dic53—28no—
186C4×Dic53—28no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53C22×D411yescharacteristic
223C23×C53C22×D411yescharacteristic
224C23×C53C2×#4111yescharacteristic
225C23×C53C2×#4111yescharacteristic
226C23×C53C2×#4112yes—
227C23×C53C2×#4111yescharacteristic
228C23×C53C2×#4112yes—
229C23×C53C2×#4111yescharacteristic
230C23×C53C2×#4111yescharacteristic
231C23×C53H(5,𝔽2)12yes—
232C23×C53H(5,𝔽2)12yes—
233D4×C53—416no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53#12314yes—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—24no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53#12114yes—
292#15120—531nomaximal, Sylow
293C22×C53—44no—
294C22×C53—44no—
295C22×C53—44no—
296C22×C53—44no—
297C22×C53—44no—
298C22×C53—44no—
299C22×C53—44no—
300C22×C53—82no—
301C22×C53—82no—
302C22×C53—82no—
303C22×C53—82no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53—82no—
307C22×C53—82no—
308C22×C53C2×#12111yescharacteristic
309C22×C53C2×#12311yescharacteristic
310C22×C53#52511yescharacteristic
311C22×C53#52511yescharacteristic
312C22×C53#52512yes—
313C22×C53#52912yes—
314C22×C53#52911yescharacteristic
315C22×C53#52911yescharacteristic
316C22×C53#53211yescharacteristic
317C22×C53#53211yescharacteristic
318C22×C53#53211yescharacteristic
319C22×C53#53511yescharacteristic
320C22×C53#53512yes—
321C22×C53#53511yescharacteristic
322C22×C53#54012yes—
323C22×C53#55112yes—
324C22×C53#55112yes—
325C22×C53#55112yes—
326C22×C53#55112yes—
327C22×C53#55312yes—
328C22×C53#55312yes—
329C22×C53#55512yes—
330C22×C53#55512yes—
331C4×C53—24no—
332C4×C53—42no—
333C4×C53—42no—
334C4×C53—82no—
335Dic53—24no—
336Dic53—44no—
337Dic53—42no—
338Dic53—42no—
339Dic53—42no—
340Dic53—82no—
341C2×#390—531no—
342C2×#388—532no—
343C2×#376—531no—
344#1444—531no—
345#1442—531no—
346#1440—532no—
347#1439—532no—
348#1438—531no—
349#1438D5311yescharacteristic
350#1437—532no—
351#1434—531no—
352C2×C53—84no—
353C2×C53#239211yescharacteristic
354C2×C53#248211yescharacteristic
355C2×C53#250011yescharacteristic
356C2×C53#252612yes—
357C2×C53#253112yes—
358C2×C53#259811yescharacteristic
359C2×C53#260312yes—
360C2×C53#261012yes—
361C2×C53#261011yescharacteristic
362C2×C53#261911yescharacteristic
363C2×C53#285411yescharacteristic
364C23×D4C2×D5311yescharacteristic
365C22×C4⋊C4—532no—
366C22×C4⋊C4—531no—
367C22×C22⋊C4—532no—
368C22×C22⋊C4—532no—
369C22×C22⋊C4—532no—
370C22×C22⋊C4—532no—
371C22×C22⋊C4C2×D5312yes—
372C22×C22⋊C4C2×D5312yes—
373C22×C42—531no—
374C22×C42—531no—
375C22×C42—531no—
376#390—1064no—
377#388—1068no—
378#376—1064no—
379C2×#95—532no—
380C2×#95—531no—
381C2×#95—532no—
382C2×#95—531no—
383C2×#95—531no—
384C2×#95—531no—
385C2×#95—532no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95C2×D5311yescharacteristic
389C2×#95C2×D5311yescharacteristic
390C53#1512011yescharacteristic, complemented, Sylow
391C25C22×D5312yes—
392C22×D4—24no—
393C22×D4—22no—
394C22×D4—22no—
395C22×D4—22no—
396C22×D4—42no—
397C23×C4—531no—
398C23×C4—532no—
399C23×C4—531no—
400C23×C4—531no—
401C23×C4—531no—
402C23×C4—532no—
403C23×C4C22×D5312yes—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—2122no—
410C2×C4⋊C4—2122no—
411C2×C4⋊C4—2122no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—42no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1064no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—2122no—
428C2×C22⋊C4—2122no—
429C2×C22⋊C4—2122no—
430C2×C42—534no—
431C2×C42—534no—
432C2×C42—1062no—
433C2×C42—1062no—
434C2×C42—1062no—
435C2×C42—1062no—
436C2×C42—1062no—
437C2×C42—1062no—
438C2×C42—1062no—
439#95—22no—
440#95—22no—
441#95—1064no—
442#95—1062no—
443#95—1062no—
444#95—1064no—
445#95—1064no—
446#95—1062no—
447#95—1062no—
448#95—2122no—
449#95—2122no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24—22no—
461C24—22no—
462C24—22no—
463C24—22no—
464C24#1171994714yes—
465C24C23×D5311yescharacteristic
466C2×D4—28no—
467C2×D4—48no—
468C2×D4—48no—
469C2×D4—44no—
470C2×D4—44no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—532no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1061no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1061no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4#1171994512yes—
525C22×C4D4×D5312yes—
526C22×C4D4×D5312yes—
527C22×C4#1171994712yes—
528C22×C4#1171994712yes—
529C4⋊C4—2128no—
530C4⋊C4—2124no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—10616no—
534C22⋊C4—2128no—
535C22⋊C4—2128no—
536C22⋊C4—2128no—
537C42—1068no—
538C42—1068no—
539C42—1068no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C2×D4×D5311yescharacteristic
576C23C2×D4×D5311yescharacteristic
577C23#41×D5311yescharacteristic
578C23#41×D5312yes—
579C23#41×D5311yescharacteristic
580C23C2×#1171994712yes—
581C23C2×#1171994711yescharacteristic
582C23C2×#1171994711yescharacteristic
583C23C2×#1171994511yescharacteristic
584C23#1313326212yes—
585C23#1313326212yes—
586D4—416no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—534no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1064no—
616C2×C4—1064no—
617C2×C4—1064no—
618C2×C4—1062no—
619C2×C4—1062no—
620C2×C4—1062no—
621C2×C4—1064no—
622C2×C4—1064no—
623C2×C4—1062no—
624C2×C4—1062no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2121no—
638C2×C4—2122no—
639C2×C4—2122no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4—2122no—
643C2×C4—2121no—
644C2×C4#1313319914yes—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#121×D5311yescharacteristic
661C22C2×#1313319911yescharacteristic
662C22#1681112911yescharacteristic
663C22#1681113312yes—
664C22#1681113311yescharacteristic
665C22#1681115012yes—
666C22#1681115211yescharacteristic
667C22#1681115511yescharacteristic
668C22#1681117011yescharacteristic
669C22#1681117111yescharacteristic
670C22#1681117211yescharacteristic
671C22#1681118611yescharacteristic
672C22#1681118812yes—
673C22#1681118811yescharacteristic
674C22#1681122312yes—
675C22#1681129912yes—
676C22#1681130112yes—
677C22#1681130412yes—
678C22#1681130412yes—
679C22#1681131612yes—
680C22#1681132312yes—
681C22#1681133712yes—
682C22#1681133712yes—
683C4—24no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—1064no—
688C4—4242no—
689C4—2124no—
690C4—2122no—
691C4—2122no—
692C4—2122no—
693C2—84no—
694C2#2027795411yescharacteristic
695C2#2027898611yescharacteristic
696C2#2027917611yescharacteristic
697C2#2027949112yes—
698C2#2027955512yes—
699C2#2028027411yescharacteristic
700C2#2028033912yes—
701C2#2028039512yes—
702C2#2028040511yescharacteristic
703C2#2028046411yescharacteristic
704C2#2028304511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1438C2
prime radicalC23×D4×C53C22
hypercenter#1438D53
hypocenterC53#15120
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1438

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16810517C2semidirect
3. C2×#16810468C2semidirect
4. #20272979C2semidirect
5. #20272972C2semidirect
6. #20272964C2semidirect
7. #20272958C2semidirect
9. #20272952C2semidirect
10. #20272939C2semidirect
69. C2×C4×Dic53D4semidirect
390. C53#15120semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168105262
C2×#168105172
C2×#168104682
#202729792
#202729722
#202729642
#202729582
#202729562
#202729522
#202729392
C53×#14382
#1512053